坎宁安函数

Cunningham function Maple animation

坎宁安函数又称为皮尔逊-坎宁安函数（Pearson-Cunningham function)是英国数学家坎宁安在1908年首先研究的特殊函数,，定义如下：

${\omega }_{m,n}(x)=\frac{e^{-x+\pi i(m/2-n)}}{\mathrm{\Gamma }(1+n-m/2)}U(m/2-n,1+m,x).$

其中U为特里科米函数。

坎宁安在是在用多變數擴展的埃奇沃斯級數，依機率密度函數的矩來近似機率密度函數時用到坎宁安函数，坎宁安函数和一維或多維常係數的擴散方程有關

坎宁安函数是下列微分方程的解

$x{X}^{″}+(x+1+m)X^{\prime }+(n+\frac{1}{2}m+1)X.$

与其他函数的关系

• ${\omega }_{m,n}(x)=\frac{exp(-x+(1/2*I)*\pi *m-I*\pi *n)*\mathrm{\Gamma }(m)*HeunB(-2*m,0,2+4*n,0,\sqrt{(}x))}{\mathrm{\Gamma }(1+n-(1/2)*m)*x^m*\mathrm{\Gamma }((1/2)*m-n)}$

$+\frac{exp(-x+(1/2*I)*Pi*m-I*\pi *n)*\mathrm{\Gamma }(-m)*HeunB(2*m,0,2+4*n,0,\sqrt{(}x))}{\mathrm{\Gamma }(1+n-(1/2)*m)*\mathrm{\Gamma }(-(1/2)*m-n)}$

• ${\omega }_{m,n}=\frac{WhittakerM(0,-1/2,-x+I*\pi *((1/2)*m-n))*exp(-(1/2)*x+(1/2*I)*\pi *((1/2)*m-n))*WhittakerW(1/2+n,(1/2)*m,x)*exp((1/2)*x)}{\mathrm{\Gamma }(1+n-(1/2)*m)*x^{(1/2+(1/2)*m)}}$

级数展开

${\omega }_{0.5,0.5}(x)=(1/80640)*(120960*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*\sqrt{(}x)-141120*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*x^{(}3/2)+77616*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*x^{(}5/2)-27720*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*x^{(}7/2)+7315*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*x^{(}9/2)+(141120*I)*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*x^{(}3/2)+(27720*I)*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*x^{(}7/2)-(100800*I)*\pi *x-(7315*I)*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*x^{(}9/2)-(77616*I)*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*x^{(}5/2)-40320*\pi +(75600*I)*\pi *x^2+100800*\pi *x+(40320*I)*\pi -75600*\pi *x^2-(120960*I)*\sqrt{(}2)*\mathrm{\Gamma }(3/4{)}^2*\sqrt{(}x)+32760*\pi *x^3-(32760*I)*\pi *x^3-9945*\pi *x^4+(9945*I)*\pi *x^4+80640*{\pi }^{(}3/2)*O(x^{(}9/2))*\sqrt{(}x))/({\pi }^{(}3/2)*\sqrt{(}x))$

参考文献

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